JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
The equation \(x^2-4 x+[x]+3=x[x]\), where \([x]\) denotes the greatest integer function, has:
- A exactly two solutions in \((-\infty, \infty)\)
- B no solution
- C a unique solution in \((-\infty, 1)\)
- D a unique solution in \((-\infty, \infty)\)
Answer & Solution
Correct Answer
(D) a unique solution in \((-\infty, \infty)\)
Step-by-step Solution
Detailed explanation
\(x^2-4 x+[x]+3=x[x]\) \(\Rightarrow x^2-4 x+3=x[x]-[x]\) \(\Rightarrow(x-1)(x-3)=[x] .(x-1)\) \(\Rightarrow x=1 \text { or } x-3=[x]\) \(\Rightarrow x-[x]=3\) \(\Rightarrow\{x\}=3 \text { (Not Possible) }\) Only one solution \(x=1\) in \((-\infty, \infty)\)
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